Bruhat canonical form for linear systems

Based on a generalization of the classical Bruhat factorization of nonsingular matrices to arbitrary rectangular matrices, a new canonical form for state space equivalence of controllable and observable linear systems is introduced. The proposed canonical form is shown to be closely related to a canonical form due to Bosgra and van der Weiden. Moreover, in the single-input single-output case and up to minor details, the proposed normal form is equivalent to the continued fraction canonical form introduced by Kalman. Connections to a cell decomposition by Fuhrmann and Krishnaprasad are discussed as well. Discrete invariants appearing in the Bruhat canonical form are shown to be invariants for restricted output feedback equivalence.